Subsets

A subset is a set whose elements all belong to another set.

The diagram above is called a Venn diagram.

More formally, a set A is a subset of a set B if every element of A is also an element of B.

When this happens, we say that A is contained in B, or that A is included in B.

This relationship is known as inclusion.

The symbol used to indicate inclusion is ⊆.

It is read as "A is a subset of B" or "A is contained in B".

The notation may also be read as "A is included in B".

The inclusion relation can be expressed in the following equivalent form:

Note. A subset is not necessarily smaller than the set that contains it. The inclusion relation also applies when A and B are exactly the same set. This is reflected in the symbol ⊆, which combines the subset symbol (⊂) with the equality sign (=).

Proper Subset

A set A is a proper subset of a set B if every element of A belongs to B and B contains at least one element that does not belong to A.

In other words, A is contained in B, but the two sets are not equal.

$$ A \ne B $$

This type of relationship is known as proper inclusion or strict inclusion.

It is usually denoted by the symbol ⊂.

It is read as "A is a proper subset of B".

You may also read it as "A is strictly included in B".

Example

Let

$$ A = \{1,3,4\} $$

$$ B = \{1,2,3,4,6,7\} $$

Every element of A belongs to B, but B contains additional elements that are not in A.

Therefore:

$$ A \subset B $$

Inclusion vs. Proper Inclusion
A proper inclusion relationship (A⊂B) always implies that the two sets are different. An inclusion relationship (A⊆B), on the other hand, allows the possibility that the two sets are equal. Therefore:

$$ A \subset B \Longrightarrow A \subseteq B $$

However, the converse is not necessarily true because A and B may contain exactly the same elements.

Improper Subset

Every set has two improper subsets:


• the set itself
• the empty set

A] The Set Itself

Every set is a subset of itself.

If two sets are equal, each is a subset of the other.

$$ A = B $$

This happens because every element of A belongs to B, and every element of B belongs to A.

In this situation:

• A is an improper subset of B
• B is an improper subset of A

Mutual inclusion is one of the standard ways to prove that two sets are equal.

Example

Consider the sets

$$ A = \{1,3,4\} $$

$$ B = \{1,3,4\} $$

Every element of A belongs to B, and every element of B belongs to A.

Therefore:

$$ A = B $$

B] The Empty Set

The empty set is a subset of every set.

The empty set is the unique set that contains no elements.

It is denoted by the symbol Ø.

$$ \varnothing = \{\} $$

Note. Although this result may seem surprising at first, it follows directly from the definition of a subset. Since the empty set contains no elements, there is no element that can violate the subset condition.

Example

Let

$$ \varnothing = \{\} $$

$$ A = \{1,3,4\} $$

Then:

$$ \varnothing \subset A $$

The inclusion is proper because A contains at least one element, whereas the empty set contains none.

Proof. Assume, for contradiction, that the empty set is not a subset of A. Then there must exist at least one element of the empty set that does not belong to A. Since the empty set has no elements, such an element cannot exist. This contradiction proves that the empty set is a subset of A.

Alternative proof. The union of a set A with any subset B⊆A is equal to A. $$ A \cup B = A $$ The same property holds for the empty set: $$ A \cup \varnothing = A $$ For example: $$ \{1,3,4\} \cup \varnothing = \{1,3,4\} $$ Since the union leaves A unchanged, the empty set behaves exactly like a subset of A.

Singleton

What is a singleton?

A singleton is a set containing exactly one element.

Example

Consider the sets

$$ A = \{1,2,3,4,5\} $$

$$ B = \{2\} $$

The set B contains only one element, so it is a singleton.

Since B is also contained in A, it is a singleton subset of A.

$$ B \subset A $$

Difference Between Inclusion and Membership

One of the most common mistakes in elementary set theory is confusing inclusion with membership.

• Inclusion (⊆) is a relation between two sets. It indicates that all the elements of one set belong to another set. $$ A \subseteq B $$
• Membership (∈) is a relation between an element and a set. It indicates that the element belongs to the set. $$ a \in A $$

Note. Similar distinctions apply to non-membership and non-inclusion:

$$ a \notin A $$

$$ A \nsubseteq B $$

Notes

Here are some additional facts about subsets.

• The Number of Subsets of a Set with n Elements
  A set with n elements has exactly \(2^n\) subsets. Each element can either be included in a subset or excluded from it, giving two independent choices for every element.

Example. A set with three elements has \(2^3 = 8\) subsets. A set with four elements has \(2^4 = 16\) subsets.